Computational materials science

Electronic structure methods solve the Schrödinger equation to calculate the energy of a system of electrons and atoms, the fundamental units of condensed matter. Many variations of electronic structure methods exist of varying computational complexity, with a range of trade-offs between speed and accuracy.

Due to its balance of computational cost and predictive capability density functional theory (DFT) has the most significant use in materials science. DFT most often refers to the calculation of the lowest energy state of the system; however, molecular dynamics (atomic motion through time) can be run with DFT computing forces between atoms.

While DFT and many other electronic structures methods are described as ab initio, there are still approximations and inputs. Within DFT there are increasingly complex, accurate, and slow approximations underlying the simulation because the exact exchange-correlation functional is not known. The simplest model is the Local-density approximation (LDA), becoming more complex with the generalized-gradient approximation (GGA) and beyond. An additional common approximation is to use a pseudopotential in place of core electrons, significantly speeding up simulations.

This section discusses the two major atomic simulation methods in materials science. Other particle-based methods include material point method and particle-in-cell, most often used for solid mechanics and plasma physics, respectively.

Molecular dynamics (MD) is the name for simulating atomic motion through time. Interactions between atoms are defined and fit to both experimental and electronic structure data with a wide variety of models, called interatomic potentials. With those interactions (forces), Newtonian motion is numerically integrated. The simplest models include only van der Waals type attractions and steep repulsion to keep atoms apart. Increasingly complex models include charges (e.g. ceramics), bonds and angles (e.g. polymers), approximations of electronic effects (e.g. metals), and more. Some models use fixed bonds, defined at the start of the simulation, while others have dynamic bonding. Most recent efforts strive for robust, transferable models with generic functional forms: spherical harmonics, Gaussian kernels, and neural networks. In addition, MD can be used to simulate groupings of atoms within generic particles, called coarse-grained modeling, e.g. creating one particle per monomer within a polymer.

Monte Carlo in the context of materials science most often refers to atomistic simulations relying on rates. In kinetic Monte Carlo (kMC) rates for all possible changes within the system are defined and probabilistically evaluated. Because there is no restriction of directly integrating motion (as in molecular dynamics), kMC methods are able to simulate significantly different problems with much longer timescales.

The methods listed here are among the most common and the most directly tied to materials science specifically, where atomistic and electronic structure calculations are also widely used in computational chemistry and computational biology and continuum level simulations are common in a wide array of computational science application domains.

Other methods within materials science include cellular automata for solidification and grain growth, Potts model approaches for grain evolution and other Monte Carlo techniques, as well as direct simulation of grain structures analogous to dislocation dynamics.

Dislocations are crystalline defects in materials with line type character. Rather than simulating the full atomic detail, discrete dislocation dynamics (DDD) directly simulates line objects. Through the theories and equations of plasticity, DDD moves dislocations through time and defines the rules describing how dislocations interact when they cross.

There are other methods for simulating dislocation motion, from full molecular dynamics simulations, continuum dislocation dynamics, and phase field models.

Phase field methods are focused on phenomena dependent on interfaces and interfacial motion. Both the free energy function and the kinetics (mobilities) are defined in order to propagate the interfaces within the system through time.

Crystal plasticity simulates the effects of atomic-based, dislocation motion without directly resolving either. Instead, the crystal orientations are updated through time with elasticity theory, plasticity through yield surfaces, and hardening laws. In this way, the stress-strain behavior of a material can be determined.

Finite element methods divide systems in space and solve the relevant physical equations throughout that decomposition. This ranges from thermal, mechanical, electromagnetic, to other physical phenomena. It is important to note from a materials science perspective that continuum methods generally ignore material heterogeneity and assume local materials properties to be identical throughout the system.

Phase diagrams are integral to materials science and the development computational phase diagrams stands as one of the most important and successful examples of ICME. The Calculation of PHase Diagram (CALPHAD) method does not generally speaking constitute a simulation, but the models and optimizations instead result in phase diagrams to predict phase stability, extremely useful in materials design and materials process optimization.

Concurrent simulations in this context means methods used directly together, within the same code, with the same time step, and with direct mapping between the respective fundamental units.

One type of concurrent multiscale simulation is quantum mechanics/molecular mechanics (QM/MM). This involves running a small portion (often a molecule or protein of interest) with a more accurate electronic structure calculation and surrounding it with a larger region of fast running, less accurate classical molecular dynamics. Many other methods exist, such as atomistic-continuum simulations, similar to QM/MM except using molecular dynamics and the finite element method as the fine (high-fidelity) and coarse (low-fidelity), respectively.[2]

Hierarchical simulation refers to those which directly exchange information between methods, but are run in separate codes, with differences in length and/or time scales handled through statistical or interpolative techniques.

A common method of accounting for crystal orientation effects together with geometry embeds crystal plasticity within finite element simulations.[2]

Building a materials model at one scale often requires information from another, lower scale. Some examples are included here.

The most common scenario for classical molecular dynamics simulations is to develop the interatomic model directly using density functional theory, most often electronic structure calculations. Classical MD can therefore be considered a hierarchical multi-scale technique, as well as a coarse-grained method (ignoring electrons). Similarly, coarse grained molecular dynamics are reduced or simplified particle simulations directly trained from all-atom MD simulations. These particles can represent anything from carbon-hydrogen pseudo-atoms, entire polymer monomers, to powder particles.

Density functional theory is also often used to train and develop CALPHAD-based phase diagrams.